Soft-Constrained Scenario RBDO: Complexity-based Modulation of Failure Probability Bounds

Codes to prun the analysis **run MAIN_RBDO_withSoftScenarioConstraints_Extended.m

Program-1 (soft-constrained joint reliability requirements)

 % min_{d\in \Theta , \zeta^{(i)}>0} \ lbrace J(d) +\rho \sum\limits_{i=1}^{N}  \zeta^{(i)}
 Such that: w(d,\delta{(i)} \leq \zeta^{(i)} \rbrace

 where
 \delta are the available scenarios (samples of the uncertain factors)
 d\in \Theta is a design vector (e.g. fitting coefficients, tunable parameters etc) in a convex design set \Theta
 J(d) is a convex cost function
 $\rho$ is a parameter weighting the cost of violating constraints and
 w(d,\delta)=\max\limits_{j\in\{1,..,n_g \}| g_j(d,\delta) % w is a convex worst-case reliability performance function,
 n_g  is the number of individual reliability requirements defined by the performance functions g_j j=1,...,n_g

Program-2 (soft-constrained individual reliability requirements)

% min_{d\in \Theta , \zeta_j^{(i)}>0} \ lbrace J(d) +\sum\limits_{j=1}^{n_g} \rho_j \sum\limits_{i=1}^{N}  \zeta_j^{(i)}
% Such that: g_j(d,\delta) \leq \zeta_j^{(i)} i=1,...,N,~j=1,..,n_g\rbrace
% where $\rho_j$ are parameters weighting the cost of violation on the reliability requirement g_j

% for this probelm the support scenarios (complexity is S_N*)
% S_N^*= the number of active constriaints + the number of violating constraints

Reliability bounds via Scenario optimization theory

Pobability[Pf(d*)<\epsilon]>1-\beta
where \beta is a small confidence parameter selected by the analyst and
\epsilon is a reliabiity bounds provided by scenario theory
fixing a confidence \epsilon is a function f(N,\beta,sN^*)
where N is the number of samples in program 1 and 2 and
s_N^* start is the complexity of the solution

References

convex scenario program with relaxed constraints

Bounds derived and applied in:

• [] Garatti, Simone & Campi, Marco. (2019). Risk and complexity in scenario optimization. Mathematical Programming. https://doi.org/10.1007/s10107-019-01446-4
• [] Roberto Rocchetta, Luis G. Crespo, A scenario optimization approach to reliability-based and risk-based design: Soft-constrained modulation of failure probability bounds, Reliability Engineering & System Safety, 2021, -.

non-convex Scenario RBDO

• [] Roberto Rocchetta, Luis G. Crespo, Sean P. Kenny, A scenario optimization approach to reliability-based design, Reliability Engineering & System Safety, Volume 196, 2020, 106755, ISSN 0951-8320, https://doi.org/10.1016/j.ress.2019.106755.